[Paper Review] Piecewise Analytic Subactions for Analytic Dynamics

Gonzalo Contreras, Artur O. Lopes+2 more|arXiv (Cornell University)|Apr 22, 2009

Mathematical Dynamics and Fractals26 references3 citations

TL;DR

This paper investigates the asymptotic behavior of eigenfunctions φ_β for the Ruelle operator associated with analytic potentials β log g under piecewise analytic expanding maps. Under uniqueness and genericity conditions, it proves that the limit 1/β log φ_β as β → ∞ is a calibrated subaction that is piecewise analytic, though not necessarily analytic everywhere, with analyticity depending on the support of the maximizing probability measure.

ABSTRACT

We consider a piecewise analytic expanding map f: [0,1]-> [0,1] of degree d which preserves orientation, and an analytic positive potential g: [0,1] -> R. We address the analysis of the following problem: for a given analytic potential beta log g, where beta is a real constant, it is well known that there exists a real analytic (with a complex analytic extension to a small complex neighborhood of [0,1]) eigenfunction phi_beta for the Ruelle operator. One can ask: what happen with the function phi_beta, when beta goes to infinity. The domain of analyticity can change with beta. The correct question should be: is 1/ beta log phi_beta analytic in the limit, when beta goes to infinity ? Under a uniqueness assumption, this limit, when beta goes to infinity, is in fact a calibrated subaction V (see bellow definition). We show here that under certain conditions and for a certain class of generic potentials this continuous function is piecewise analytic (but not analytic). In a few examples one can get that the subaction is analytic (we need at least to assume that the maximizing probability has support in a unique fixed point).

Motivation & Objective

• To understand the limiting behavior of eigenfunctions φ_β for the Ruelle operator as β → ∞.
• To determine whether the rescaled logarithm 1/β log φ_β converges to a calibrated subaction V.
• To investigate the regularity of the limiting subaction V under analytic potential and expanding map conditions.
• To establish conditions under which V is piecewise analytic rather than fully analytic.
• To explore the role of the maximizing probability's support in determining analyticity of the subaction.

Proposed method

• Analyzes the Ruelle operator acting on analytic functions for piecewise analytic expanding maps of degree d on [0,1].
• Considers analytic positive potentials g and the associated Ruelle operator eigenfunctions φ_β for β log g.
• Applies complex analytic extension techniques to study φ_β in a complex neighborhood of [0,1].
• Uses a uniqueness assumption to ensure convergence of 1/β log φ_β to a unique calibrated subaction V.
• Employs techniques from thermodynamic formalism and subaction theory to analyze the regularity of V.
• Examines the structure of V by analyzing the support of the maximizing probability measure.

Experimental results

Research questions

• RQ1 Does the limit 1/β log φ_β exist and converge to a calibrated subaction as β → ∞?
• RQ2 Under what conditions is the limiting subaction V piecewise analytic rather than analytic?
• RQ3 How does the support of the maximizing probability influence the analyticity of the subaction V?
• RQ4 Can V be analytic even when the maximizing measure has support on multiple points?
• RQ5 What role does the genericity of the potential play in determining the regularity of V?

Key findings

• The limit 1/β log φ_β converges to a calibrated subaction V as β → ∞ under a uniqueness assumption.
• The limiting subaction V is piecewise analytic, but not necessarily analytic across the entire interval [0,1].
• Analyticity of V is guaranteed only when the maximizing probability has support on a single fixed point.
• For generic potentials, the subaction V exhibits a piecewise analytic structure due to discontinuities in the derivative at certain points.
• The domain of analyticity of φ_β may change with β, but the limiting behavior stabilizes into a well-defined subaction V.
• The subaction V is not analytic in general, even for analytic potentials, unless the maximizing measure is supported on a unique fixed point.